The Asymmetric Effects of Uncertainty on Inflation and Output Growth

Transcription

1 The Asymmeric Effecs of Uncerainy on Inflaion and Oupu Growh Kevin B. Grier Deparmen of Economics, Universiy of Oklahoma & Ólan T. Henry, Nilss Olekalns and Kalvinder Shields Deparmen of Economics, The Universiy of Melbourne This Version: Augus 23, 2002 Absrac We sudy he effecs of growh volailiy and inflaion volailiy on average raes of oupu growh and inflaion for poswar U.S. daa. Our resuls sugges ha growh uncerainy is associaed wih higher average growh and lower average inflaion. Inflaion uncerainy is significanly negaively correlaed wih boh oupu growh and average inflaion. Boh inflaion and growh display evidence of significan asymmeric response o posiive and negaive shocks of equal magniude. Keywords: Growh; Inflaion; Uncerainy; Asymmery; Generalised Impulse Response Funcions; J.E.L. Numbers: E390 The auhors hank Chris Brooks, Paul Cashin, Rodolfo Cermeno, Tim Dunne, Robin Grier, and paricipans a he Sixh Annual Ausralian Macroeconomics Workshop and he 2002 Ausralasian Meeing of he Economeric Sociey for heir commens on preliminary drafs of his paper. Of course, any errors are our responsibiliy.

2 1 Inroducion Quesions regarding he relaionship beween inflaion and real aciviy are fundamenal empirical issues in macroeconomics. Does uncerainy abou growh promoe or reard growh? Is he effec of inflaion uncerainy pernicious? Do growh and inflaion respond asymmerically o posiive and negaive shocks of equal magniude? Recenly, much aenion has been focussed on relaionships beween uncerainy abou inflaion and growh and heir average oucomes, see Grier and Perry (1998, 2000), Ramey and Ramey (1995) and Henry and Olekalns (2002) iner alia. Researchers have used a variey of approaches o measure uncerainy. However, he grea majoriy of empirical work is eiher univariae, or else uses resricive models of he covariance process. Univariae models by definiion do no allow sudy of he join deerminaion of he wo series, and popular covariance-resriced mulivariae models can be subjec o severe specificaion error, see Kroner & Ng (1998). In his paper we specify and esimae an exremely general model of oupu growh and inflaion. Unlike he previous research, our model allows for he possibiliies of spillovers and asymmeries in he variance covariance srucure for inflaion and growh. The resuls show ha our model provides a superior condiional daa characerizaion o he resriced approaches previously employed in he lieraure. We also employ simulaion mehods o highligh he economic imporance of hese sources of non-lineariy in he daa. The paper is organized as follows. Secion 2 describes our daa and he esing process we use o parameerise our model. In secion 3 we repor esimaion resuls and diagnosic ess for model adequacy and discuss he implicaions of our resuls for. 1

3 several well-known heories of he effecs of uncerainy on inflaion and oupu growh. The fourh secion discusses he quaniaive effecs of uncerainy in he model along wih he naure of he asymmeric effecs of inflaion and oupu growh shocks on uncerainy. The final secion summarises our conclusions. 2. Economeric Model and Daa Descripion The daa used in his sudy are for he US, and were obained from he FRED daabase a he Federal Reserve Bank of Sain Louis. The sample is monhly daa over he period April 1947 o Ocober We measure inflaion, π, as he annualized, monhly difference of he logarihm of he producer price index. Similarly we measure oupu growh, y, as he annualized, monhly difference of he logarihm of he index of indusrial producion. These daa are shown in Figure 1, and summary saisics for hese daa are presened in Table 1. -Figure 1 abou here- - Table 1 abou here - Boh oupu growh and inflaion are posiively skewed and display significan amouns of excess kurosis wih boh series failing o saisfy he null hypohesis of he Bera-Jarque (1980) es for normaliy. A baery of augmened Dickey-Fuller uni roo ess, Dickey and Fuller (1979) and Kwiakowski, Phillips, Schmid and Shin (1992) ess for saionariy sugges ha boh are I(0) series. However a series of Ljung-Box (1979) ess for serial correlaion suggess ha here is a significan amoun of serial dependence in he daa. Similarly a Ljung-Box es for serial correlaion in he squared daa provides srong evidence of condiional heeroscedasiciy in he daa. Visual inspecion of he ime series plos of he daa in. 2

4 Figure 1 would end o suppor he view ha he variances of oupu growh and inflaion are no consan. Equaion 1 gives he specificaion we use for he means of inflaion ( π ) and oupu growh ( y ). I is a VARMA (vecor auoregressive moving average), GARCH in Mean model, where he condiional sandard deviaions of oupu growh and inflaion are included as explanaory variables in each equaion: p Y = µ + Γ Y +Ψ h + Θ ε + ε i i j j i= 1 j= 1 H ( 0, H ) hy, hy π, = hy π, h π, q ε (1) where Y y ε y, y, = ; ε ; h π = ε = π, h π, h µ y ; µ = ; µ π Γ Γ = () i () i Γ i () i () i Γ21 Γ22 ; ψ Ψ= ψ ψ ψ θ ( j) ( j) θ and Θ j =. ( j) ( j) θ21 θ22 Under he assumpion ε Ω ~ (0, H), where Ω represens he informaion se available a ime, he model may be esimaed using Maximum Likelihood mehods, subjec o he requiremen ha H, he condiional covariance marix, be posiive definie for all values of ε in he sample. We use he conceps of good and bad news o inroduce an asymmery ino he condiional variance-covariance process. 2 Specifically, if inflaion is higher han expeced, we ake ha o be bad news. In his case, he inflaion residual will be 1 We choose he values of p and q ha minimize he Akaike and Schwarz informaion crieria. In he resuls below, p=q=2. 2 As a preliminary es, we subjec each of he wo series o an Engle & Ng (1993) es for asymmery in volailiy, finding ha oupu growh does exhibi negaive sign and size bias while inflaion exhibis posiive size bias. Thus here is iniial indicaive evidence ha allowing for asymmery may be imporan and ha macroeconomic bad news maers more han good news.. 3

5 posiive. By conras if oupu growh is lower han expeced, we consider ha o be bad news. Thus bad news abou oupu growh is capured by a negaive residual. We herefore define ξ y, as min{ ε y,,0} which capures he negaive innovaions, or bad news abou growh. Similarly le ξ π, be he max{ ε π,,0} (i.e. he posiive inflaion residuals), hus capuring bad news abou inflaion. We allow for asymmeric responses using (2) H = C C + A ε ε A + B H B + D ξ ξ D (2) *' * *' ' * *' * *' ' * where * * * * * * * c11 c 12 * α11 α 12 * β11 β 12 C0 = ; * A11 = ; B * * 11 = ; * * D 0 c22 α11 α22 β11 β22 δ δ = * * * * * δ 21 δ ξ y, and ξ = 2 ξπ,. 3 The symmeric BEKK model (Engle and Kroner 1995) is a special case of (2) for δ ij = 0, for all values of i and j. The BEKK parameerisaion guaranees H posiive definie for all values of ε in he sample. Diagonaliy and symmery resricions should be esed raher han, as is ofen he case, imposed since he invalid imposiion of he resricion creaes a poenially serious specificaion error. Our covariance model allows for he innovaions of inflaion and oupu growh o have boh non-diagonal and asymmeric effecs on he condiional variances of each series and he condiional covariance. The model ness simpler diagonal and symmeric models and we can provide a saisical es of heir appropriaeness. 4 3 Brooks and Henry (2000), and Brooks Henry and Persand (2002) have used his model. 4 Kroner & Ng (1998) review he properies of many widely used mulivariae GARCH models. The BEKK model does allow for non-diagonaliy, commonly imposed on he model using he resricion * α = β =0 for i,j=1,2 and i j in equaion (2) above. Some popular mulivariae covariance models ij * ij. 4

6 The wo exising papers closes o ours are Grier & Perry (2000) and Henry & Olekalns (2001). Grier & Perry examine monhly US daa using a resriced covariance model ha we show can be rejeced by he daa. Henry & Olekalns esimae an asymmeric univariae GARCH-M model for quarerly US oupu growh. This univariae approach does no allow inflaion (oupu growh) residuals o influence he condiional variance of oupu growh (inflaion), an assumpion ha is also rejeced by he daa. 3 Resuls Table 2 repors parameer esimaes for he full model given by equaions (1) and (2) above. Preliminary resuls sugges ha he assumpion of normally disribued sandardised innovaions, z, = ε, / h,, for k = y, π, may be enuous. We hus k k k follow Weiss (1986) and Bollerslev and Wooldridge (1992) who argue ha asympoically valid inference regarding normal quasi-maximum likelihood esimaes may be based upon robusified versions of he sandard es saisics. 5 - Table 2 abou here A. Specificaion ess In his secion, we consider ess on he form of he condiional covariance and he adequacy of he specificaion. Firs, here is significan condiional heeroskedasiciy in hese daa. Homoskedasiciy requires he A, B and D * * * coefficien marices o be joinly insigifican, and hey are joinly and individually significan a he 0.01 level. also impose furher resricions on he diagonal model such as he consan correlaion model of Bollerlsev (1990).. 5

7 Second, he hypohesis of a diagonal covariance process requires he offdiagonal elemens of he same hree coefficien marices o be joinly insignifican and hese esimaed coefficiens are joinly significan a he 0.05 level or beer. To be more specific, he insignificance of he non-diagonal coefficiens in he * A 11 marix indicaes ha allowing for non-diagonaliy does no increase he persisence of he condiional variances. However, he significance of he analogous coefficiens in he * B 11 and * D 11 marices, shows ha he lagged squared innovaions in each series do impac he condiional variance of he oher series in some manner. Third, he hypohesis of a symmeric covariance process requires he coefficien marix * D 11 o be insignifican. In our model, all elemens save * δ 12 are individually significan, and he overall coefficien marix is significan, a he 0.01 level. In paricular, he significance of α * 22 coupled wih he significance of δ * 22 indicaes ha inflaion displays own variance asymmery, implying ha, ceeris paribus, a posiive inflaion innovaion leads o more inflaion volailiy han a negaive innovaion of equal magniude. In a similar manner, he fac ha boh * and δ 11 are significan suggess ha, ceeris paribus, he response of oupu growh displays own variance asymmery; negaive growh shocks raise growh uncerainy more han posiive shocks. * α 11 In sum, for hese US poswar daa, he inflaion oupu growh process hus is srongly condiionally heerskedasic, innovaions o inflaion (oupu growh) significanly influcence he condiional variance of oupu growh (inflaion) and he sign, as well as he size, of boh inflaion and growh innovaions are imporan. 5 Maximum likelihood esimaion assuming a condiional Sudens- disribuion was also performed. The resuls were qualiaively unchanged. Deails are available from he second auhor upon reques.. 6

8 Overall, he model appears o be well specified. The sandardised residuals, and heir corresponding squares, saisfy he null of no fourh order linear dependence of he Q(4) and Q 2 (4) ess. Similarly here is no evidence, a he 5% level, of welfh 2 order serial dependence in, and z,. 6 We also subjec he sandardized residuals o z k k a series of ess based on momen condiions. In a well-specified model Ez ( ) = 0 k, and 2 Ez ( ) = 1. These condiions are suppored a any level of significance. The k, model also significanly reduces he degree of skewness and kurosis in he sandardised residuals when compared wih he raw daa. Similarly he model predics ha E 2 ( εk, ) = hk, for k = y, π and E( π ) ε ε = h. These condiions are no y,, yπ, rejeced by he daa a he 0.05 level. - Figure 2 abou here - In Figure 2, we plo he respecive condiional variances for he raes of inflaion and oupu growh, as well as he condiional covariance, implied by our esimaes. For oupu growh, volailiy appears highes, on average, during he 1950s. The well-documened decline in oupu growh volailiy over he 1990s is also apparen in hese daa. For inflaion, he period of greaes volailiy occurs in he mid-1970s, wih he mos benign volailiy oucomes coming during he 1960s and mid 1990s. B. Theoreical Implicaions The Ψ marix in (1) capures he relaionship beween he elemens of he sae vecor and he condiional second momens. The coefficiens of he Ψ marix can be inerpreed as he response of growh (inflaion) o he condiional variances of growh and inflaion. 6 There is some evidence of welfh order dependence in he squared sandardised residuals of inflaion.. 7

9 Do increases in growh volailiy lower, raise or have no impac on average growh? The sign and significance of ψ 11, he upper lef elemen of he Ψ coefficien marix can be used o discriminae beween hese conflicing views. This coefficien is posiive and significan a all usual confidence levels wih an asympoic -saisic of around We hus find srong evidence in favor of he correlaion implied by Fisher Black s (1987) ideas abou echnological adopion or he effecs of uncerainy on opimal saving. The predicion ha increased oupu volailiy lowers growh is no suppored in hese daa. 7 Wheher or no inflaion uncerainy lowers growh, can be deermined by he sign and significance of ψ 12. This coefficien is negaive and again significan a all usual levels wih a -saisic of over We hus find consisency wih he argumens of Friedman (1977) and Okun (1971) regarding he pernicious real effecs of inflaion uncerainy. Does higher inflaion volailiy lower raher han raise average inflaion? Cukierman (1992), and Cukierman & Melzer (1986) show ha if he money supply process has a sochasic elemen and he public is uncerain abou he objecive funcion of he policymaker, hen a sraegic policy maker will reac o an increase in uncerainy abou he supply process by raising he average level of inflaion. The relevan coefficien for he heory ha he Fed reacs o increased inflaion uncerainy by raising he average inflaion rae is ψ 22. This coefficien is negaive and 7 Previous work esing his hypohesis is exremely mixed. Using cross-counry daa, Ramey & Ramey (1995) find a significan negaive relaionship beween he sandard deviaion of growh and average growh, while Kormendi & Meguire (1985) and Grier & Tullock (1989) find a significan posiive relaionship. Using a univariae GARCH model on US daa, Caporale & McKiernan (1998) find a posiive effec, while Henry & Olekalns (2001) find a negaive relaion using an asymmeric univariae GARCH model. Grier & Perry (2000) find no effec in a symmeric bivariae GARCH model of inflaion and oupu growh, and Dawson & Sephenson (1997) reach he same conclusion from an examinaion of sae level daa.. 8

10 significan a he 0.01 level, indicaing ha higher inflaion uncerainy is associaed wih lower, raher han higher, average inflaion. 8 Finally, wha is he effec of an increase in growh volailiy on average inflaion? The predicion ha increased growh uncerainy raises average inflaion, as in Deveraux (1989), receives no suppor from he daa as can be seen from he negaive, bu small and only marginally significan coefficien of ψ Generalised Impulse Response Analysis The parameer esimaes and residual diagnosics repored above esablish he saisical significance of he asymmeric response of he condiional variancecovariance srucure o posiive and negaive shocks o growh and inflaion. We furher esablish he saisical significance of inflaion and growh volailiy for explaining he behavior of average inflaion and growh. In his secion, we (i) quanify he dynamic response of growh and inflaion o shocks and (ii) assess he economic imporance of he asymmery in he variance covariance srucure. We use Generalised Impulse Response Funcions (GIRFs), inroduced by Koop e al (1996), o analyse he ime profile of he effecs of shocks on he fuure behaviour of he growh rae and inflaion. Shocks impac on growh and inflaion 8 In a series of univariae models for each of he G7 counries, Grier & Perry (1998) find he same resul. They argue ha if higher inflaion raises uncerainy, a sabilizing Fed would reac o increased uncerainy by lowering inflaion. They found a similar resul for he UK and Germany, and found resuls consisen wih he models of Cukierman and Melzer for Japan and France. Holland (1995) also finds ha increased inflaion uncerainy lowers average inflaion in US daa, using a survey based uncerainy measure. 9 To see he imporance of allowing for non-diagonal and asymmeric responses of uncerainy o innovaions, i is insrucive o compare he above resuls wih hose in Grier & Perry (2000) who invesigae similar hypoheses using a bivariae GARCH-M model wih diagonaliy and symmery resricions. They oo find ha higher inflaion uncerainy lowers growh, bu he res of heir GARCH-M coefficiens are insignifican. By relaxing heir resricions we find srong suppor for he hypohesis ha real uncerainy and average growh are posiively correlaed and ha inflaion uncerainy and average inflaion are negaively correlaed.. 9

11 direcly hrough he condiional mean as described in (1) and wih a lag hrough he condiional variance (2). The firs advanage of using GIRFs over radiional impulse response funcions in his conex is ha hey allow for composiion dependence in mulivariae models (see also Lee and Pesaran (1993) and Pesaran and Shin (1998)), i.e. he effec of a shock o oupu growh is no isolaed from having a conemporaneous impac on inflaion and vice versa. Secondly, hey are also applicable o non-linear mulivariae models since hey avoid problems of dependence on he size, sign and hisory of he shock In more deail, if Y is a random vecor, he GIRF for a specific shock υ and hisory ω 1 is defined as GIRF ( n, υ, ω ) = E[ Y υ, ω ] E[ Y ω 1], (3) Y 1 + n 1 + n for n = 0, 1, 2, Hence, he GIRF is condiional on υ and ω 1 and consrucs he response by averaging ou fuure shocks given he pas and presen. Given his, a naural reference poin for he impulse response funcion is he condiional expecaion of Y + given only he hisory ω 1, and, in his benchmark response, he n curren shock is also averaged ou. Assuming ha υ and ω 1 are realisaions of he random variables V and Ω ha generae realisaions of { }, hen, following Koop 1 e al (1996), he GIRF defined in (3) can be considered o be a realisaion of a random variable given by, GIRFY( n, V, Ω 1) = E[ Y+ n V, Ω 1] E[ Y+ n Ω 1 ]. (4) Y The compuaion of GIRFs for non-linear models is made difficul by he inabiliy o consruc analyical expressions for he condiional expecaions. Mone Carlo mehods of sochasic simulaion, herefore, need o be used o compue he. 10

12 condiional expecaions (see Granger and Teräsvira (1993, Ch. 8), and Koop e al (1996) for deailed descripions of he various mehods ha can be used). The GIRFs for our esimaed model are shown in Figures 3 hrough 6. Figure 3 shows he effec on growh of an iniial uni sized growh rae shock. The GIRF is consisen wih he growh rae iniially declining afer he impac of he shock. Then, afer he firs quarer, here is a simulus in he growh rae (peaking a a 0.5 percenage poin of he iniial uni shock afer 6 monhs), which akes approximaely hree years o fully dissipae. Figures 3,4,5 & 6 abou here A growh shock has a much more persisen impac on he inflaion rae, alhough he magniude of his effec is very small. The relevan GIRF is shown in Figure 4. Four years afer he shock, inflaion is only around 0.04 percenage poins higher han if he shock had no occurred. Even a is peak, a around 24 monhs, he effec on inflaion of a growh rae shock is small. Figures 5 and 6 relae o a uni shock o he inflaion rae. Wih respec o he growh rae, an inflaion shock firs provides a large simulus o growh bu hen he growh rae falls afer around 6 monhs. In Figure 6, inflaion quickly falls afer he iniial impac of he inflaion shock. The impac, however, is reasonably persisen; afer four years, inflaion is around 0.4 of a percenage poin higher han i would have been oherwise. 10 Given he asymmeric naure of he model specificaion, one use of he GIRFs is in he evaluaion of he significance of any asymmeric effecs of posiive and negaive growh and inflaion shocks on boh oupu growh and inflaion. For insance, he response funcions can be used o measure he exen o which negaive. 11

13 shocks may be more persisen han posiive shocks as well as assess he poenial diversiy in he dynamics in he effecs of posiive and negaive shocks on oupu growh and inflaion. Le GIRFY( n, V +, Ω 1) denoe he GIRF derived from + condiioning on he se of all possible posiive shocks, where V = { υ υ > 0} and GIRF ( n, V +, Ω ) Y 1 denoe he GIRF from condiioning on he se of all possible negaive shocks. The disribuion of he random asymmery measure, ASY ( n, V, Ω ) = GIRF ( n, V, Ω ) + GIRF ( n, V, Ω 1 ) (5) Y 1 Y 1 Y will be zero if posiive and negaive shocks have exacly he same effec. Hence he disribuion can provide an indicaion of he asymmeric effecs of posiive and negaive shocks (van Dijk e al. 2000). Compuaion of he asymmery measures for a growh (inflaion) shock o he growh and inflaion series sugges he following. Firs, all four measures show saisical significance alhough hey vary in relaive magniudes. Second, a negaive oupu shock o oupu growh and inflaion gives more persisence (on average) relaive o he corresponding posiive shock. For insance, he asymmery measure for a growh shock o growh is , wih a -raio of , and he asymmery measure for a growh shock o inflaion is wih a -raio of Third, he response of boh oupu growh and inflaion o a posiive inflaion shock shows a more persisen effec relaive o a negaive inflaion shock. The respecive asymmery measures for an inflaion shock o growh and inflaion are (wih -raio equal o 5.491) and (wih -raio equal o 2.855). 10 All he GIRF s are precisely esimaed where he impulse responses in (i) Figure 5 are significanly differen from zero up unil he 33 rd monh, and in (ii) Figures 6, 7 and 8 are all significanly differen from zero for he ime horizon shown (50 monhs).. 12

14 5 Conclusions The resuls in he paper imply ha virually all exising ARCH or GARCH models of inflaion or oupu growh are misspecified and herefore are suspec wih regard o heir inferences. We have shown ha for he Unied Saes, he condiional volailiies of inflaion and oupu growh exhibi significan non-diagonaliy and asymmery wih respec o he impac of lagged innovaions. Volailiy in one series spills over ino volailiy in he oher, and he size and sign of he innovaion (our disincion beween good and bad news) has a differenial impac upon he esimaed condiional variance-covariance marix. We find srong evidence in favor of he proposiion ha growh uncerainy is associaed wih a higher average rae of growh. We find no evidence ha increased growh uncerainy increases he average rae of inflaion. On he oher hand, inflaion uncerainy is associaed wih lower average growh raes. Conrary o he predicion ha inflaion uncerainy induces policymakers o raise he average inflaion rae, we find ha inflaion uncerainy is associaed wih lower average inflaion raes. We use simulaion mehods o highligh he impac and persisence of shocks o growh and inflaion on fuure growh and inflaion. These simulaions emphasise he economically significan effecs of he asymmeric response of variancecovariance srucure of growh and inflaion o news.. 13

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18 Table 1: Summary Saisics Mean Variance Skewness Excess Kurosis Bera-Jarque Normaliy Y π Uni Roo and Saionariy Tess ADF(τ) ADF(µ) ADF KPSS(µ) KPSS(τ) Y π % C.V Tess for Serial Correlaion and ARCH Q(4) Q(12) Q 2 (4) Q 2 (12) ARCH(4) Y π Noes o Table 1: Marginal significance levels displayed as [.]. 17

19 Table 2: The Mulivariae Asymmeric GARCH-in-Mean model Condiional Mean Equaions p Y = µ + Γ Y +Ψ h + Θ ε + ε Y i i j j i= 1 j= 1 q y µ Γ Γ ψ ψ ; i i = ; µ = ; Γ i = ; i i Ψ = π µ 2 Γ ψ 21 Γ22 21 ψ22 h j j y, ε y, θ11 θ 12 h = ; ε = ; j j j h ε Θ = π, θ π, 21 θ ( ) (0.0121) (0.0102) ˆ µ = Γ ˆ 1 = ( ) ( ) ( ) ˆ Θ 1 = (0.0243) (0.0467) ( ) ( ˆ Θ 2 = ˆ Γ 2 = (0.0117) (0.0109) (0.0045) (0.0046) (0.0274) (0.0559) ) ( ) ( ) (0.0065) ( ) Ψ= ˆ ( ) ( ) Residual Diagnosics Mean Variance Q(4) Q 2 (4) Q(12) Q 2 (12) ε , [0.7225] [0.9969] [0.5764] [0.1885] [0.0446] [0.4622] ε , [0.5035] [0.9991] [0.7474] [0.2298] [0.4924] [0.0078] Momen Based Tess 2 2 E( ε y, ) = hy, E( ε π, ) = hπ, E( εε ) = h y, π, yπ, [0.4267] [0.0574] [0.1561] Noes: Sandard errors displayed as (.). Marginal significance levels displayed as [.]. Q(m) and Q 2 (m) are are Ljung-Box ess for m h 2 order serial correlaion in z k, and zk, respecively for k =y,π.. 18

20 Table 2 Coninued: Esimaes of he Mulivariae Asymmeric GARCH Model Condiional Variance-Covariance Srucure *' * *' ' * *' * *' ' * H = C C + A ε ε A + B H B + D ξ ξ D εy, 1 min( εy, 1,0) ε 1 = ; ξ 1 ε = π, 1 max( επ, 1,0) (0.0817) (0.1595) (0.0026) (0.0213) ˆ * C0 = * Bˆ 11 = (0.0977) ( ) (0.0064) (0.0255) (0.0139) (0.0147) (0.0176) ˆ * * A 11 = Dˆ 11 = ( ) (0.0179) ( ) (0.0518) Diagonal VARMA i i i i H0 : Γ 12 =Γ 21 = θ12 = θ21 = 0 No GARCH-M H0 : ψ = 0 for all i, j ij No asymmery: H 0 :δ ij =0 for i,j=1,2 Diagonal GARCH * * * * * * H : α = α = β = β = δ = δ =

21 80 Oupu Growh Inflaion Figure 1: The Daa. 20

22 35 Condiional Sandard Deviaion: Oupu Condiional Sandard Deviaion: Inflaion Inflaion-Oupu Covariance Figure 2: Esimaed Condiional Sandard Deviaions and Condiional Covariance. 21

23 Figure 3: GIRF Shock o Growh on Growh Figure 4: GIRF Shock o Growh on Inflaion. 22

24 Figure 5: GIRF Shock o Inflaion on Growh Figure 6: GIRF Shock o Inflaion on Inflaion. 23